Ratio test

In mathematics, the ratio test is a test (or "criterion") for the convergence of a series where each term is a real or complex number and an is nonzero when n is large.

More specifically, let Then the ratio test states that:[2][3] If the limit L in (1) exists, we must have L = R = r. So the original ratio test is a weaker version of the refined one.

This illustrates that when L = 1, the series may converge or diverge: the ratio test is inconclusive.

In such cases, more refined tests are required to determine convergence or divergence.

Below is a proof of the validity of the generalized ratio test.

has infinite non-zero members, otherwise the series is just a finite sum hence it converges.

Similar to the above case, we may find a natural number

Extensions to the ratio test, however, sometimes allow one to deal with this case.

[4][5][6][7][8][9][10][11] In all the tests below one assumes that Σan is a sum with positive an.

These tests also may be applied to any series with a finite number of negative terms.

Any such series may be written as: where aN is the highest-indexed negative term.

The first expression on the right is a partial sum which will be finite, and so the convergence of the entire series will be determined by the convergence properties of the second expression on the right, which may be re-indexed to form a series of all positive terms beginning at n=1.

Each test defines a test parameter (ρn) which specifies the behavior of that parameter needed to establish convergence or divergence.

All of the tests have regions in which they fail to describe the convergence properties of Σan.

Accordingly, there will be no distinction drawn between references which use one or the other form of the test parameter.

This extension is due to Joseph Ludwig Raabe.

The proof of the other half is entirely analogous, with most of the inequalities simply reversed.

This extension is due to Joseph Bertrand and Augustus De Morgan.

[4][9][13] The series will: This extension probably appeared at the first time by Margaret Martin in 1941.

[14] A short proof based on Kummer's test and without technical assumptions (such as existence of the limits, for example) was provided by Vyacheslav Abramov in 2019.

is large, can be presented in the form (The empty sum is assumed to be 0.

The series For applications of Extended Bertrand's test see birth–death process.

This extension is due to Carl Friedrich Gauss.

Assuming an > 0 and r > 1, if a bounded sequence Cn can be found such that for all n:[5][7][9][10] then the series will: This extension is due to Ernst Kummer.

Let ζn be an auxiliary sequence of positive constants.

Then, Hence, Note that for these four tests, the higher they are in the De Morgan hierarchy, the more slowly the

is monotonically decreasing and positive which in particular implies that it is bounded below by 0.

Therefore, the limit This implies that the positive telescoping series and since for all

The provided modification of Kummer's theorem characterizes all positive series, and the convergence or divergence can be formulated in the form of two necessary and sufficient conditions, one for convergence and another for divergence.

Another ratio test that can be set in the framework of Kummer's theorem was presented by Orrin Frink[19] 1948.